On the stability problem for conditional expectation

The behavior of the conditional expectation Es{;Xvb;Ys}; under a small perturbation Z of the conditioning random variable Y is analyzed. We show that if Y and Z are independent then Es{;Xvb;Y + [var epsilon]Zs}; converges to Es{;Xvb;Ys}; in mean as [var epsilon] --> 0 for all integrable X, provided the distribution of Y is absolutely continuous. We also show that the limit is Es{;Xvb;Y, Zs}; rather than Es{;Xvb;Ys};, i.e., there is no stability, when Y is a discrete (i.e., countably valued) random variable. Finally, we show that in general Es{;Xvb;Y + [var epsilon]Zs}; might have no limit in distribution as [var epsilon] --> 0.